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24,2
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High order Nédélec elements with
local complete sequence
properties
374
Joachim Schöberl and Sabine Zaglmayr
Institute for Computational Mathematics, Johannes Kepler University Linz,
Austria
Abstract
Purpose – The goal of the presented work is the efficient computation of Maxwell boundary and
eigenvalue problems using high order H(curl) finite elements.
Design/methodology/approach – Discusses a systematic strategy for the realization of arbitrary
order hierarchic H(curl)-conforming finite elements for triangular and tetrahedral element geometries.
The shape functions are classified as lowest order Nédélec, higher-order edge-based, face-based (only
in 3D) and element-based ones.
Findings – Our new shape functions provide not only the global complete sequence property but also
local complete sequence properties for each edge-, face-, and element-block. This local property allows
an arbitrary variable choice of the polynomial degree for each edge, face, and element. A second
advantage of this construction is that simple block-diagonal preconditioning gets efficient. Our high
order shape functions contain gradient shape functions explicitly. In the case of a magnetostatic
boundary value problem, the gradient basis functions can be skipped, which reduces the problem size,
and improves the condition number.
Originality/value – Successfully applies the new high order elements for a 3D magnetostatic
boundary value problem, and a Maxwell eigenvalue problem showing severe edge and corner
singularities.
Keywords Finite element analysis, Eigenvalues, Eigenfunctions
Paper type Research paper
1. Introduction
Electromagnetic problems are formulated in the function space
H ðcurlÞ :¼ {u [ ½L 2 ðVÞ3 : curl u [ ½L 2 3 }:
COMPEL: The International Journal
for Computation and Mathematics in
Electrical and Electronic Engineering
Vol. 24 No. 2, 2005
pp. 374-384
q Emerald Group Publishing Limited
0332-1649
DOI 10.1108/03321640510586015
It naturally contains the continuity of the tangential components across sub-domain
interfaces. This property has led to the construction of finite elements with tangential
continuity. The most prominent one is the edge element, which is the lowest order
member of the first family of Nédélec elements (Nédélec, 1980). A recent monograph is
in Monk (2003). Up from the lowest order element all of them contain edge-based
degrees of freedom. Higher order ones also contain unknowns in the element faces, and
in the interior of the elements. To match the functions over the element-interfaces, the
orientation of the edges and faces has to be considered. While on edges this is just
changing the sign, the orientation of triangular faces is more involved. In Webb (1999),
the problem was solved by defining rotational invariant sets of basis functions for
Both authors are supported by the Austrian Science Foundation FWF under project grant
Start-project Y 192 “hpFEM : Solvers and Adaptivity”.
the various orders. In Ainsworth and Coyle (2003) rotational symmetry was given up,
which resulted in several reference elements.
Together with H 1 :¼ {u [ L 2 : 7 u [ ½L 2 3 } and H ðdivÞ :¼ {u [ ½L 2 3 :
div u [ L 2 }; the function spaces form the sequence
Curl
7
High order
Nédélec elements
div
H 1 ! H ðcurlÞ ! H ðdivÞ ! L 2 :
375
Moreover, this sequence is complete, i.e.
rangeð7Þ ¼ kerðcurlÞ;
rangeðcurlÞ ¼ kerðdivÞ:
This complete sequence property is inherited on the finite element level, if
continuous elements ðW h;pþ1 Þ; Nédélec elements ðV h;p Þ; Raviart-Thomas elements
ðQh;p21 Þ; and discontinuous elements ðS h;p22 Þ of proper polynomial order are
chosen for the four spaces, respectively. This relation is visualized in the De Rham
complex:
H1
curl
7
div
! H ðcurlÞ ! H ðdivÞ !
<
<
7
W h;pþ1 !
V h;p
<
curl
L2
<
div
! Qh;p21 ! S h;p22 :
The complete sequence property is essential for the convergence of the finite
element approximation, in particular for eigenvalue problems (Monk et al., 2001;
Boffi, 2000). Also the kernel-preserving multigrid preconditioners of Arnold et al.
(2000) and Hiptmair (1999) are based on the knowledge of the curl-free functions.
In this paper, we construct arbitrary order basis functions allowing individual
polynomial orders for each edge, face, and element of the mesh. The resulting finite
element spaces fulfill the complete sequence property. We show that for our new basis
functions, block-Jacobi preconditioners are kernel preserving and thus effective.
Numerical experiments show the moderate growth of the condition number for
increasing polynomial order. An extended version of the present paper is in
preparation (Schöberl and Zaglmayr, 2005).
2. Hierarchical high-order shape functions
We construct hierarchical high-order basis functions on triangular and tetrahedral
elements. The extension to all other common element geometries is in Schöberl and
Zaglmayr (2005). We start with the construction for H 1 finite elements, and derive the
corresponding H(curl) elements (Figures 1 –4).
Hierarchical finite elements are usually based on orthogonal polynomials.
Let ð‘i Þi¼0; ... ;p denote the Legendre-polynomials up to order p. This is a basis
for P p ð½21; 1Þ: Furthermore, define the integrated Legendre-polynomials ðLi Þi¼2; ... ;p
via
Z x
Li ðxÞ :¼
‘i21 ðsÞ ds:
21
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376
Figure 1.
Figure 2.
High order
Nédélec elements
377
Figure 3.
They form a basis for P p0 ð½21; 1Þ :¼ {c [ P p ð½21; 1Þ : cð21Þ ¼ cð1Þ ¼ 0}: The
edges of the element are Em, where m ¼ 1; 2; 3 for triangles, and m ¼ 1; . . . ; 6 for
tetrahedra. An edge E m has two vertices denoted by ðe1 ; e2 Þ: We assume that all edges
are oriented such that e1 , e2 : A tetrahedron has four faces denoted by F 1 ; . . . ; F 4 :
A face is represented by three vertex indices ðf 1 ; f 2 ; f 3 Þ: We assume that they are
ordered such that f 1 , f 2 , f 3 :
The basis functions are expressed in terms of the barycentric coordinates l1, l2, and
l3, and l4 for the 3D element. Following Karniadakis and Sherwin (1999), we exploit a
tensor product structure on the simplicial elements.
We introduce the scaled Legendre polynomials
s
n
‘S
ðs;
tÞ
:¼
t
‘
n
n
t
and scaled integrated Legendre polynomials
n
LS
n ðs; tÞ :¼ t Ln
s
t
on the triangular domain with vertices (2 1, 1), (1, 1), (0, 0)
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378
Figure 4.
t [ ð0; 1; s [ ½2t; t:
Both can be evaluated by division-free three term recurrences. We observe that LS
n
vanishes on the two edges s ¼ 2t and s ¼ t; and corresponds to Ln(s) on the third edge
t ¼ 1: By means of two barycentric coordinates li ; lj $ 0; li þ lj # 1; this domain
can be parameterised as
ðs; tÞ ¼ ðli 2 lj ; li þ lj Þ:
2.1 High2 order scalar shape functions
We start with presenting hierarchical high order finite elements for approximating
scalar H 1 functions. These functions must be continuous over element boundaries.
This means that the point values in element vertices, and the polynomials along edges
(and faces in 3D) must be the same. This is obtained by defining basis functions
associated with vertices, edges, (faces), and the interior of elements. The vertex basis
functions are the standard linear nodal functions being one in one vertex, and
vanishing in the other ones. The edge based functions have to span P p0 on the edge, and
must vanish at the other two edges. The scaled integrated Legendre polynomials fulfil
this property. In 2D, the interior basis functions must vanish on the boundary of the
triangle. They are defined as tensor product of 1D basis functions. One factor has to
vanish on two edges, the second one has to eliminate the third edge. In 3D, the face
shape functions are defined by means of scaled Legendre polynomials, which ensures
that the argument takes values on the whole interval (2 1, 1), and thus improves the
conditioning.
2.2 High-order vector shape functions
Next, we explain our new vector-valued basis functions. The basic idea is to
include the gradients of the scalar basis functions into the set of basis functions. If
the scalar function is continuous, then the gradient has continuous tangential
components. First, we take the lowest order Nédélec elements, which have one
basis function for each edge. The additionally needed edge-based basis functions
for spanning P p on the edge can be taken exactly as the gradients of the scalar
edge-based functions which span P 0pþ1 : Since the edge-based functions are
gradients, the curl of them vanishes. The interior basis functions for the triangle
need vanishing tangential components. This is obtained by taking the gradients of
the H 1-interior functions. But now, the gradient fields are not enough to span all
vector-valued functions. Recall that the scalar basis is built as tensor product uivj.
The evaluation of the gradient gives 7ðui vj Þ ¼ ð7ui Þvj þ ui 7vj : Note that not only
the sum, but both individual terms have vanishing tangential boundary values.
Thus, also different linear combinations like ð7ui Þvj 2 ui 7vj can be taken as basis
functions. By counting the dimension one finds that still p2 1 functions are
missing. These can be chosen linearly independent as the product of the
edge-element function, and a polynomial vj. By means of the scaled Legendre
polynomials, the 2D construction is easily extended to 3D elements.
2.3 Local complete sequence properties
The global scalar finite element space can be decomposed into vertex, edge, (face,) and
element based spaces:
High order
Nédélec elements
379
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380
W pþ1 ¼ W V þ
X
X
X
W Ei þ
W Fi þ
W Ti , H 1
Ei
Fi
Ti
Similarly, the H(Curl) finite element space is split into the edge-element basis, the
high-order edge, (face) and element blocks:
X
X
X
V Ei þ
V Fi þ
V T i , H ðCurlÞ:
V p ¼ V N0 þ
Ei
Fi
Ti
Similarly, one defines high order basis functions for H(div) as lowest order
Raviart-Thomas Q RT0, high order face, and high order element-based functions,
and the L2 spaces as constants, and high order element functions. We constructed
the basis functions such that each one of the blocks satisfies a local complete
sequence property:
7
curl
div
0
0
! QRT
! S h;0
W Vh;pþ1¼1 ! V N
h
h
7
W EpE þ1 ! V EpE
7
curl
7
! QIpI 21 ! S IpI 22
W FpF þ1 ! V FpF ! QFpF 21
W IpI þ1 ! V Ipl
curl
div
Some of the advantages of this local property are the following.
.
An arbitrary individual polynomial order can be chosen on each edge, face
and element. The global complete sequence property is satisfied
automatically.
.
Simple edge-face-element block-diagonal preconditioning becomes efficient.
.
The basis functions contain high order gradient fields explicitly. A simple way of
gauging (as needed, e.g. for the magnetostatic boundary value problem) can be
performed by skipping the high order gradient fields. Note that the lowest order
gradient fields are still in the system.
.
The implementation of the gradient-operator from H 1 to H(curl) finite element
spaces becomes trivial.
This kind of basis-functions has been implemented in our software package NgSolve
for common 2D and 3D element geometries (triangles, quadrilaterals, tetrahedra,
prisms and hexahedra).
3. Magnetostatic boundary value problem
We consider the magnetostatic boundary value problem
curlm 21 curl A ¼ j;
where j is the given current density, and A is the vector potential for the magnetic flux
B ¼ curl A: For gauging, we add a small (six orders of magnitude smaller than m 2 1)
zero-order term to the operator. In order to show the performance of the constructed
shape functions we compute the magnetic field induced by a cylindrical coil.
The mesh generated by Netgen (Schöberl, n.d.) contains 2,035 curved tetrahedral
elements. Figures 5 and 6 show the magnetic fieldlines and the absolute value of the
magnetic flux simulated with H(curl)-elements of uniform order six.
We have chosen different polynomial orders p, and compared the number of
unknowns (dofs), the condition numbers of the preconditioned system, the required
iteration numbers of the PCG for an error reduction by 102 10, and the required
computation time on an Acer notebook with Pentium Centrino 1,600 MHz CPU. We run
the experiments once with keeping the gradient shape functions, and a second time
with skipping them. The computed B-field is the same for both versions. One can
observe a considerable improvement of the required solver time in the Table I.
High order
Nédélec elements
381
3.1 Maxwell Eigenvalue problem
We consider the Maxwell eigenvalue problem: find v – 0 such that
curl E ¼ ivmH ;
curl H ¼ 2iv1E:
The corresponding weak form is to find eigenvalues v . 0 and eigenvectors E [
H ðcurlÞ; E – 0 such that there holds
Z
Z
m 21 curl E · curl v dx ¼ v 2
1E · v dx
V
V
for all v [ H ðcurlÞ: Finite element discretization leads to the generalized matrix
eigenvalue problem
Figure 5.
Magnetic field induced by
the coil, order p ¼ 6
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382
Figure 6.
jcurl Aj ¼ jBj; order p ¼ 6
Table I.
Performance of the solver
p
dofs
2
2
3
3
4
4
5
5
6
6
7
19719
10686
50884
29130
104520
61862
186731
112952
303625
186470
286486
Grads
yes
no
yes
no
yes
no
yes
no
yes
no
no
k(C 2 1A)
Iter
Solvertime (s)
7.9
7.9
24.2
18.2
71.4
32.3
179.9
55.5
421.0
84.0
120.0
20
21
32
31
48
40
69
49
97
59
68
1.9
0.7
9.8
2.9
40.5
10.7
137.9
31.9
427.8
87.4
209.6
Au ¼ v 2 Mu:
It is well known that this system contains many zero-eigenvalues which correspond to
the gradient fields. A standard eigenvalue solver such as the inverse power iteration
would suffer from first computing all the unwanted zero-eigenvalues. Following
Hiptmair and Neymeyr (2002), we perform an inexact inverse iteration with inexact
projection: Given un, we compute
ln ¼
ðAun ; un Þ
ðMun ; un Þ
High order
Nédélec elements
u~ nþ1 ¼ un 2 C 21
AþsM ðAun 2 ln Mun Þ
~ un
unþ1 ¼ ðI 2 PÞ~
383
C 21
AþsM
Here,
is a preconditioner for the shifted H(curl) problem, and I2 P̃ is an inexact
projection into the complement of gradient fields. It is realized by performing k steps of
the inexact projection
T
k
~ nþ1
unþ1 ¼ ðI 2 B7 C 21
D B7 M V Þ u
Here, B7 is the matrix representing the gradient, MV is the mass matrix for the
vector-elements, and CD is a Poisson-preconditioner. As mentioned above, the gradient
operator is very simple for the presented basis functions.
We have chosen to compute the Neumann-eigenvalues on the Fichera domain
½21; 13 n½0; 13 : It shows severe singularities along the non-convex edges and at the
vertex in the origin. We have chosen a priori a mesh refinement with three levels of
anisotropic hp-refinement to resolve the singularities. Figure 7 shows the first
non-trivial eigenvector approximated by elements of order p ¼ 4:
Figure 7.
First Maxwell eigenvector
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